Scaling invariance for the diffusion coefficient in a billiard system

TL;DR

This work analyzes unbounded diffusion in a time-dependent oval billiard and its suppression by inelastic boundary collisions. It derives a diffusion-coefficient $D$ and solves the associated diffusion equation to reveal a scale-invariant structure across control parameters $(\epsilon,\eta)$ and dissipation $\gamma$, including a short-time constant $D$, a crossover, and a power-law decay with $\beta=-1$. Both phenomenological and analytical arguments establish a generalized homogeneous-function description that yields consistent scaling exponents and a universal collapse under rescaling, linking chaotic billiard dynamics with diffusion theory. The results clarify how dissipation constrains Fermi-like acceleration and provide a framework for applying diffusion equations to chaotic dissipative systems, with potential broader applicability in transport problems. The crossover scale and exponents indicate a universality class shared with the dissipative standard map, reinforcing the connection between nonlinear dynamics and statistical mechanics in driven, lossy systems.

Abstract

We investigated the unbounded diffusion observed in a time-dependent oval-shaped billiard and its suppression owing to inelastic collisions with the boundary. The main focus is on the behavior of the diffusion coefficient, which plays a key role in describing the scaling invariance characteristic of this transition. For short times, the low-action regime is characterized by a constant diffusion coefficient, which begins to decay after a crossover iteration, thereby suppressing the unlimited growth of velocity. We demonstrate that this behavior is scaling-invariant concerning the control parameters and can be described by a homogeneous generalized function and its associated scaling laws. The critical exponents are determined both phenomenologically and analytically, including the decay exponent beta = -1, previously identified in the diffusion coefficient of the dissipative standard map.

Figures (3)

• Figure 1: (Color online) $V_{rms}$$vs.$$n$ for (a) the conservative case with $\epsilon=0.08$, $p=3$, $\eta=0.5$ and different values of $V_0$ and (c) the dissipative case for initial velocity $V_0 = 10^{-5}$ for different values of $\gamma$ and $\eta \epsilon$. (b) and (d) present the overlap of the curves in (a) and (c), respectively, onto single universal curves using the adequate scaling transformations.
• Figure 2: Plot of curves of $D/(\epsilon\eta)^2(1+\gamma)^2$ (a) $vs. \ n$ for different control parameters and (b) $vs. \ n(1-\gamma^2)$ overlapping the curves in (a) onto a single universal plot. In both cases, the symbols represent numerical curves while the continuous lines are obtained from equation (17).
• Figure 3: (a) Plot of $D(n)$$vs.$$(1-\gamma^2)$. A power law fitting yielded an exponent of $-0.99(9)$. (b) Plot of $D(n)$$vs.$$(1-\gamma)$ with a power law fit giving $z_2=-0.99(5)$.